Mechanics of Options Markets

European and American Option

  • A European option may be exercised only at the expiration date of the option, i.e. at a single pre-defined point in time.
  • An American option on the other hand may be exercised at any time before the expiration date.

Types of Option

  • An option gives the holder of the option the right to do something. But it’s not forced.
  • Call: the right to buy something(underlying 标的, including stock/future/foreign currency)
  • Put: the right to sell.
  • Premium: the cost to buy an option.
  • expiration date, maturity date. Time limit to execution. The execution of an option is ‘exercise an option’.
  • exercise/strike price. The holder could buy/sell something at exercise/strike price.
  • It’s obvious that At expiration date:
    • underlying price > strike price, call option profits.
    • underlying price < strike price, put option profits.
  • Under an opposite situation, the holder doesn’t exercise the option. So the loss is Premium rather than the difference between the two prices.
  • Position: the number of options that a guy maintains.
    • Long Position: increase the position
    • Short Position: decrease the position
    • Also represent (buy/purchase)-(sell/write) a option.

Margin Requirements

The price of option is usually low, but the price of underlying may be high and fluctuate significantly. To ensure the investor could burden the fluctuation, investor must prepare extra money in his account, which is called margin. And the margin requirement is related to the position. More options you maintain, more margin you should have.

Equity style

  • taker: no margin requirement
  • writer: about (underlying price * 10%)

Price

  • in the money = 价内 = 实值 = call strike price < underlying price = put strike price > underlying price = profit
  • at the money = 价中 = 平值 = strike price == underlying price
  • out of the money = 价外 = 虚值 = call strike price > underlying price = put strike price <> underlying price = loss

Pricing

Syntax

syntax explanation
$S_0$ Current stock price
$K$ Strike price of option
$T$ Time to expiration of option
$S_T$ Stock price on the expiration date
$\sigma$ Volatility of the stock price
$r$ Continuously compounded risk-free rate of interest for an investment maturing in time T
$C$ Value of American call option to buy one share
$P$ Value of American put option to sell one share
$c$ Value of European call option to buy one share
$p$ Value of European put option to sell one share.

The effects.

Variable C P c p
$S_0$ + - + -
$K$ - + - +
$T$ + + ? ?
$\sigma$ + + + +
$r$ + - + -

Continuously Compounded Interest

$$\lim_{n \to +\infty} (1 + \frac{r}{n})^n = e^{r}$$

Bounds

  • $c \leq S_0 \quad and \quad C \leq S_0$
    • long a stock and short a option results in risk-free profit.
  • $c \geq S_0-Ke^{-rT}$ | short a stock and long a option.
    • Then hold cash and get interest. if stock price increases, call option will profit. Otherwise, don’t exercise the option and buy a stock in market.
  • $p \geq max(Ke^{-rT}-S_0, 0)$
    • negative put makes no sense.
  • $S_0-K \leq C-P \leq S_0-Ke^{-rT}$

Call-Put Parity

  • Portfolio A: one European call option plus a zero-coupon bond that provides a payoff of K at time T.
  • Portfolio C : one European put option plus one share of the stock.
  • At expiry time T, A is equal to C.
    • $S_T>K$:
      • A: $S_T-K+K=S_T$
      • C: $0+S_T=S_T$
    • $S_T<K$:
      • A: $0+K=K$
      • C: $K-S_T+S_T=K$
  • So the current cost should be equal.
  • $c+Ke^{-rT}=S_0+p$

Trading Strategies Involving Options

Principal-Protected

One option + One Underlying.

Spread

A spread trading strategy involves taking a position in two or more options of the same
type.

Bull Spread

buying a European call option on a stock with $K_1$ and selling a European call option with a $K_2$. The original cost is $C_d$. It should be negative because out of money option has lower price.

  • $K_1<K_2$
    • $S_0<K_1 => -C_d$
    • $K_1<S_0<K_2 => S_0 - K_1 - C_d$
    • $S_0>K_2 => K_2 - K_1 - C_d$

Three types of bull spread can be distinguished:

  1. Both calls are initially out of the money.
  2. One call is initially in the money; the other call is initially out of the money.
  3. Both calls are initially in the money
Bear Spread

buying a European put with $K_2$ and selling a European put with $K_1$. And The original cost is $P_d$.

  • $K_1<K_2$
    • $S_0<K_1 => K_2 - K_1 - P_d$
    • $K_1<S_0<K_2 => K_2 - S_0 - P_d$
    • $S_0>K_2 => -P_d$
Box Spread

a bull call spread with strike prices K1 and K2 and a bear put spread with the same two strike prices.

  • $K_1<K_2$
    • $S_0<K_1 => K_2 - K_1 - P_d - C_d$
    • $K_1<S_0<K_2 => K_2 - K_1 - P_d - C_d$
    • $S_0>K_2 => K_2 - K_1 - P_d - C_d$
Butterfly Spread

involves positions in options with three different strike prices.

Calendar Spreads

have the same strike price and different expiration dates.

Diagonal Spreads

both the expiration date and the strike price of the calls are different.

Combinations

Binomial Tree

Generalization

Follow a straightforward formula(or assumption):

$S_up+S_d(1-p)=S_0*e^{rT}$

Left is the expected value of stock price.

Right is risk-free profit.

They should be equal, if not, there is an arbitrage chance.

Some equivalent formats:

  • $uS_0p+dS_0(1-p)=S_0*e^{rT}$
  • $up+d-dp=e^rT$
  • $p = \frac{e^{rT}-d}{u-d}$

Delta

The delta ($\Delta$) of a stock option is the ratio of the change in the price of the stock
option to the change in the price of the underlying stock.

Determining u and d

  • $u = e^{\sigma\sqrt{T}}$
  • $d = \frac{1}{u}$
  • $\sigma$ is the volatility.

Black-Scholes-Merton Model

Assumptions about How stock price evolves

  • $\mu$ : Expected Return on Stock
  • $\sigma$ : Volatility of the stock price
  • $\frac{\Delta S}{S}\sim\phi(\mu\Delta t, \sigma^2\Delta t)$ (normal distribution)

Reference

Fundamentals_of_Options_and_Futures_Markets（8th_Edition）John Hull